On the relationship between the Wigner–Moyal approach and the quantum operator algebra of von Neumann

Abstract

In this paper we discuss the close relationship between the Wigner---Moyal algebra and the original non-commutative quantum algebra introduced by von Neumann in 1931. We show that the "distribution function", F(P, X, t) is simply the quantum mechanical density matrix for a single particle where the coordinates, X and P, are not the coordinates of a point particle, but the mean co-ordinate of a cell structure (a `blob') in phase space. This provides an intrinsically non-local and non-commutative description of an individual, which only becomes a point particle in the commutative limit. In this general structure, the Wigner function appears as a transition probability amplitude which accounts for the appearance of negative values. The Moyal and Baker brackets play a significant role in the time evolution, producing the quantum Hamilton---Jacobi equation used in the Bohm approach. It is the non-commutative structure based on a symplectic geometry that generates a generalised phase space for quantum processes.